The weights of weighted median and recursive weighted median filters have close relationship with the root signal characteristics of these filters. Depending on the given weights, a signal may or may not be a root signal for these filters. In this thesis, we find a set of conditions of weights under which a recursive weighted median filter preserves monotone or locally monotone sequences and that under which any input sequence converges to a locally monotone sequence after a finite number of passes. We also find a set of conditions on weights under which the output of a recursive weighted median filter is the same as that of a recursive median filter. Finally, we present several examples to show the derived weight conditions more explicitly.